Non-projected Calabi–Yau supermanifolds over $\mathbb{P}^{2}$

We start a systematic study of non-projected supermanifolds, concentrating on supermanifolds with fermionic dimension $2$ and with the reduced manifold a complex projective space. We show that all the non-projected supermanifolds of dimension $2 \vert 2$ over $\mathbb{P}^2$ are completely characterised by a non-zero cohomology class $\omega \in H^1 (\mathcal{T}_{\mathbb{P}^2} (-3))$ and by a locally free sheaf $\mathcal{F}_{\mathcal{M}}$ of rank $0 \vert 2$, satisfying $\mathcal{Sym}^2 \mathcal{F}_{\mathcal{M}} \cong K_{\mathbb{P}^2}$. Denoting such supermanifolds with $\mathbb{P}^2_\omega (\mathcal{F}_{\mathcal{M}})$, we show that all of them are Calabi–Yau supermanifolds and, when $ \omega \neq 0$, they are non-projective, that is they cannot be embedded into any projective superspace $\mathbb{P}^{n \vert m}$. Instead, we show that every non-projected supermanifold over $\mathbb{P}^2$ admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold $\mathbb{P}^2_\omega (\mathcal{F}_{\mathcal{M}})$ that cannot be embedded in any of the $\Pi$-projective superspaces $\mathbb{P}^n_{\Pi}$ introduced by Manin and Deligne. However, we also show that when $\mathcal{F}_{\mathcal{M}}$ is the cotangent bundle over $\mathbb{P}^2$, then the non-projected $\mathbb{P}^2_\omega (\mathcal{F}_{\mathcal{M}})$ and the $\Pi$-projective plane $\mathbb{P}^2_{\Pi}$ do coincide.

Full Text (PDF format)

Received 18 March 2018

Accepted 22 March 2019

Published 25 October 2019